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4. Consider the vector field F = (, y, 23). (a) By expressing all quantities in terms of spherical polar coordinates, find an expres- sion for the vector surface element dS of a sphere S of unit radius, centred on the origin, and hence evaluate the surface integral I = = F .dS. You may use the results 2 (sin*x + cos x)dr= and sin xr = (b) Evaluate the integral 12 = SSS 28 where V is the volume enclosed by the sphere S defined in part (a). (c) Evaluate the divergence of the vector field F. Hence, comment on the relationship between the values of I, and 12 and state the theorem that it demonstrates. (d) Evaluate the vector field A=curl F. Hence, trivially, evaluate 1 = S A.dS. A By considering the fact that S is a closed surface, discuss why the value of Ig is consistent with Stokes's theorem.